Understanding the Reaction of the Plane on a Weight

When a weight of 500 kg is held on a smooth plane inclined at 30° to the horizontal by a force P acting 30° above the plane, finding the reaction of the plane on the weight requires a thorough understanding of the forces involved and their application of Newton's laws. This article delves into the detailed steps and calculations to determine the reaction force.

Introduction to the Problem

A 500 kg weight is placed on a smooth plane inclined at an angle of 30° to the horizontal. A force P is exerted at an angle of 30° above the plane to keep the weight in place. Our objective is to determine the reaction force of the plane on the weight.

Step-by-Step Analysis

To solve this problem, we need to analyze the forces acting on the weight along and perpendicular to the inclined plane. This can be achieved using the principles of vector resolution and equilibrium conditions.

Weight Calculation

The weight force W can be calculated using the formula:

W mg 500 kg × 9.81 m/s2 4905 N

Resolution of Weight

The weight force can be resolved into two components:

Parallel Component: The component of the weight parallel to the plane.

Perpendicular Component: The component of the weight perpendicular to the plane.

The calculations for these components are as follows:

Parallel Component:

Wparallel W sin 30° 4905 sin 30° 4905 × 0.5 2452.5 N

Perpendicular Component:

Wperpendicular W cos 30° 4905 cos 30° ≈ 4905 × 0.866 4254.4 N

Force P Decomposition

The force P can also be resolved into two components:

Parallel Component: Pparallel P cos 30°

Perpendicular Component: Pperpendicular P sin 30°

Equilibrium Conditions

For the weight to be in equilibrium, both the parallel and perpendicular force components must balance:

Parallel Forces Equilibrium:

Pparallel Wparallel

Perpendicular Forces Equilibrium:

R - Pperpendicular Wperpendicular

Where R is the reaction force from the plane.

Calculation of Reaction Force R

By rearranging the perpendicular equilibrium condition equation, we get:

R Wperpendicular - Pperpendicular

Since the perpendicular components of the weight and force P need to balance, we can substitute:

R 4254.4 N - P sin 30°

Or, more simply:

R 4254.4 N - P/2

Conclusion

To find the exact value of the reaction force R, we need to know the value of P. By understanding the components of the forces and applying the principles of equilibrium, we can calculate the reaction force R.

Additional Considerations

For a complete analysis, it's important to consider the angle and magnitude of the force P. If provided with the value of P, you can directly substitute it into the equation to determine the reaction force R.

Understanding the forces and their components is crucial for performing such calculations accurately and efficiently.